Airy Turning-Point Asymptotics for Ramanujan's Entire Function $A_q(z)$

Abstract

Let $$ A_q(z)=\sum_{k=0}^{\infty}\frac{q^{k^2}}{(q;q)_k}(-z)^k $$ be Ramanujan's entire function. We study it in the turning-point scaling $$ q=e^{-\varepsilon},\qquad z=\frac{\sqrt q}{4}e^{-\varepsilon^{2/3}\zeta}, $$ with $\zeta$ in a compact subset of $\mathbb{C}$. After an explicit exponential normalization, a direct coalescing-saddle analysis gives, uniformly on compact subsets, the expansion $$ \operatorname{Ai}(\zeta) +\frac{\varepsilon^{2/3}}{30} \left(4\zeta\operatorname{Ai}(\zeta) +\zeta^2\operatorname{Ai}'(\zeta)\right) +O_K(\varepsilon). $$ Thus the first correction is explicit and comes with a quantitative remainder. For every fixed $n$, the same analysis locates the positive zero associated with the $n$-th Airy zero and proves that it is globally the $n$-th positive zero of $A_q$. Expanding the normalized $q$-difference equation recovers the Airy differential equation and confirms the scaling. An appendix records Morita's antisymmetric companion, whose normalized limit is $\operatorname{Bi}$, together with a single-valued meromorphic descent of it. Numerical tables illustrate the normalization, the correction, and the zero formulas.